MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovrspc2v Structured version   Visualization version   GIF version

Theorem ovrspc2v 6836
Description: If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
Assertion
Ref Expression
ovrspc2v (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑦,𝑌   𝑥,𝑋,𝑦
Allowed substitution hint:   𝑌(𝑥)

Proof of Theorem ovrspc2v
StepHypRef Expression
1 oveq1 6821 . . 3 (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
21eleq1d 2824 . 2 (𝑥 = 𝑋 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑦) ∈ 𝐶))
3 oveq2 6822 . . 3 (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
43eleq1d 2824 . 2 (𝑦 = 𝑌 → ((𝑋𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑌) ∈ 𝐶))
52, 4rspc2va 3462 1 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  (class class class)co 6814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817
This theorem is referenced by:  mgmcl  17466  mndpropd  17537  issubmnd  17539  submcl  17574  issubg2  17830  gass  17954  lmodprop2d  19147  lsspropd  19239  gsummatr01lem2  20684  off2  29773  ofcf  30495  submgmcl  42322  clcllaw  42355
  Copyright terms: Public domain W3C validator